Numerical simulation of first-order velocity-dilatation-rotation elastic wave equation with staggered grid
WANG Hui1,2, HE Bingshou1,2, SHAO Xiangqi3
1. Key Lab of Submarine Geosciences and Prospecting Techniques, MOE China, Ocean University of China, Qingdao, Shandong 266100, China; 2. Functional Laboratory of Marine Mineral Resources Evaluation and Exploration Technology, Qingdao National Laboratory of Marine Science and Technology, Qingdao, Shandong 266100, China; 3. Yantai Natural Resources and Planning Bureau, Yantai, Shandong 264003, China
Abstract:Elastic wave forward modeling plays an important role in seismic wave propagation mechanism research and the acquisition, processing, interpretation, and inversion of multi-wave seismic data. Present forward modeling of elastic wave equations often numerically solves first-order velocity-stress equation or second-order displacement equation to only obtain three particle vibration velocity components or displacement components containing P- and S-wave. The wave-field decoupling operator should be employed to separate P- and S-wave for a more intuitive recording of pure P- and S-wave components. Therefore, the accuracy of the wave records simulated by the methods is subject to the accuracy of both the simulation algorithm and the wavefield decoupling algorithm. This paper derives the higher-order finite-diffe-rence scheme of the first-order velocity-dilatation-rotation elastic wave equation in three-dimensional staggered grid space and gives the corresponding stability conditions. The PML absorbing boundary conditions adapted to the equation are derived, and the forward modeling of the first-order velocity-dilatation-rotation elastic wave equation is realized. The physical meaning of each component in the simulation results is analyzed. The equation not only contains the vibration velocity vector of the particles but also explicitly includes the P- and S-wave vibration velocity vectors. Additionally, an volumetric strain and a rotation vector are also involved. Therefore, in addition to the three particle vibration velocity components, the decoupled P- and S-wave components can be obtained directly by the equation. This avoids the influence of the decoupling algorithm on decoupling accuracy, and model trials prove he validity and superiority of the proposed method.
ALTERMAN Z,KARAL F C. Propagation of elastic waves in layered media by finite difference methods[J]. Bulletin of the Seismological Society of America,1968,58(1): 367-398.
[2]
BAYLISS A,JORDAN K E,LEMESURIER B J,et al. A fourth-order accurate finite difference scheme for the computation of elastic waves[J]. Bulletin of the Seismological Society of America,1986,76(4): 1115-1132.
[3]
IGEL H,RIOLLET B,MORA P. Accuracy of staggered 3-D finite-difference grids for anisotropic wave propagation[C]. SEG Technical Program Expanded Abstracts,1992,11: 1244-1246.
[4]
董良国,马在田,曹景忠,等. 一阶弹性波方程交错网格高阶差分解法[J]. 地球物理学报,2000,43(3): 411-419.DONG Liangguo,MA Zaitian,CAO Jingzhong,et al. A staggered-grid high-order difference method of one-order elastic wave equation[J]. Chinese Journal of Geophysics,2000,43(3): 411-419.
[5]
张会星,何兵寿,宁书年. 双相介质中纵波方程的高阶有限差分解法[J]. 物探与化探,2004,28(4): 307-309,313.ZHANG Huixing,HE Bingshou,NING Shunian. High-order finite difference solution of dilatational wave equations in two-phase media[J]. Geophysical and Geochemical Exploration,2004,28(4): 307-309,313.
[6]
CRASE E. High-order (space and time) finite-diffe-rence modeling of the elastic wave equation[C]. SEG Technical Program Expanded Abstracts,1990,9: 987-991.
[7]
HESTHOLM S,RUUD B O. 3-D finite-difference elastic wave modeling including surface topography[J]. Geophysics,1998,63(2): 613-622.
[8]
HE C,QIN G,WEI Z. High-order finite-difference modeling on reconfigurable computing platform[C]. SEG Technical Program Expanded Abstracts,2005,24: 1755-1758.
[9]
牟永光,裴正林.三维复杂介质地震数值模拟[M]. 北京: 石油工业出版社,2005.
[10]
MADARIAGA R. Dynamics of an expanding circular fault[J]. Bulletin of the Seismological Society of America,1976,66(3): 639-666.
[11]
VIRIEUX J. SH-wave propagation in heterogeneous media: Velocity-stress finite-difference method[J]. Geophysics,1984,49(11): 1933-1942.
[12]
LEVANDER A R. Fourth-order finite-difference P-SV seismograms[J]. Geophysics,1988,53(11): 1425-1436.
[13]
董良国,马在田,曹景忠. 一阶弹性波方程交错网格高阶差分解法稳定性研究[J]. 地球物理学报,2000,43(6): 856-864.DONG Liangguo,MA Zaitian,CAO Jingzhong. A study on stability of the staggered-grid high-order difference method of first-order elastic wave equation[J]. Chinese Journal of Geophysics,2000,43(6): 856-864.
[14]
孙耀充,张延腾,白超英. 二维弹性及粘弹性TTI介质中地震波场数值模拟: 四种不同网格高阶有限差分算法研究[J]. 地球物理学进展,2013,28(4): 1817-1827.SUN Yaochong,ZHANG Yanteng,BAI Chaoying. Seismic wavefield simulation in 2D elastic and visco-elastic media: comparison between four different kinds of finite-difference grids[J]. Progress in Geophysics,2013,28(4): 1817-1827.
[15]
CHARI-HYUN J,SHIN C,SUH J H. An optimal 9-point,finite-difference,frequency-space,2-D scalar wave extrapolator[J]. Geophysics,1996,61(2): 529-537.
[16]
MIN D J,SHINZ C,KWON B D,et al. Improved frequency-domain elastic wave modeling using weighted-averaging difference operators[J]. Geophysics,2000,65(3): 884-895.
[17]
FEI T,LARNER K. Elimination of numerical dispersion in finite-difference modeling and migration by flux-corrected transport[J]. Geophysics,1995,60(6): 1830-1842.
[18]
何兵寿,魏修成,刘洋. 三维波动方程的数值频散关系及其叠前和叠后数值模拟[J]. 石油大学学报(自然科学版),2001,25(1): 67-71.HE Bingshou,WEI Xiucheng,LIU Yang. Numerical dispersion relation of the three dimensional wave equation and numerical simulation of pre-stack and post-stack[J]. Journal of University of Petroleum (Edition of Natural Science),2001,25(1): 67-71.
[19]
吴国忱,王华忠. 波场模拟中的数值频散分析与校正策略[J]. 地球物理学进展,2005,20(1): 58-65.WU Guochen,WANG Huazhong. Analysis of nume-rical dispersion in wave-field simulation[J]. Progress in Geophysics,2005,20(1): 58-65.
[20]
CERJAN C,KOSLOFF D,KOSLOFF R,et al. A non-reflecting boundary condition for discrete acoustic and elastic wave equation[J]. Geophysics,1985,50(4): 705-708.
[21]
HIGDON R L. Absorbing boundary condition for elastic waves[J]. Geophysics,1991,56(2): 231-241.
[22]
BERENGER J P. A perfectly matched layer for the absorption of electromagnetic waves[J]. Journal of Computational Physics,1994,114(2): 185-200.
[23]
罗玉钦,刘财. 近似完全匹配层边界条件吸收效果分析及衰减函数的改进[J]. 石油地球物理勘探,2018,53(5): 903-913.LUO Yuqin,LIU Cai. Analysis of absorption effect and improvement of attenuation function for boundary conditions of approximately perfect matching layer[J]. Oil Geophysical Prospecting,2018,53(5): 903-913.
[24]
何兵寿,张会星,韩令贺. 弹性波方程正演的粗粒度并行算法[J]. 地球物理学进展,2010,25(2): 650-656.HE Bingshou,ZHANG Huixing,HAN Linghe. Forward modelling of elastic wave equation with coarse-grained parallel algorithm[J]. Progress in Geophy-sics,2010,25(2): 650-656.
KONDOH Y. On thoughts analysis of numerical schemes for simulation using a kernel optimum nearly-analytical discretization (KOND) method[J]. Journal of the Physical Society of Japan,1991,60(9): 2851-2861.
[27]
YANG D H,TENG J W,ZHANG Z J,et al. A nearly analytic discrete method for acoustic and elastic wave equations in anisotropic media[J]. Bulletin of the Seismological Society of America,2003,93(2): 882-890.
[28]
OPRSAL I,ZAHRDNIK J. Elastic finite-difference method for irregular grids[J]. Geophysics,1999,64(1): 240-250.
[29]
PITARKA A. 3D elastic finite-difference modeling of seismic motion using staggered grid with nonuniform spacing[J]. Bulletin of the Seismological Society of America,1999,89(1): 54-68.
[30]
DELLINGER J,ETGEN J. Wave-field separation in two-dimensional anisotropic media[J]. Geophysics,1990,55(7): 914-919.
[31]
SUN R,CHOW J,CHEN K J. Phase correction in separating P- and S-waves in elastic data[J]. Geophysics,2001,66(5): 1515-1518.
张婧,张文栋,张铁强,等. 应用τ-p域矢量旋转的地震数据波场分离[J]. 石油地球物理勘探,2020,55(1): 46-56.ZHANG Jing,ZHANG Wendong,ZHANG Tieqiang,et al. Wave field separation of seismic data using vector rotation in τ-p domain[J]. Oil Geophysical Prospecting,2020,55(1): 46-56.
[34]
何兵寿,高琨鹏,徐国浩. 各向异性介质中弹性波逆时偏移技术的研究现状与展望[J]. 石油物探,2021,60(2): 210-223.HE Bingshou,GAO Kunpeng,XU Guohao. Elastic wave reverse time migration in anisotropic media: state of the art and perspectives[J]. Geophysical Prospecting for Petroleum,2021,60(2): 210-223.
[35]
TANG H G,HE B S,MOU H B. P- and S-wave ener-gy flux density vectors[J]. Geophysics,2016,81(6): T357-T368.
[36]
李凯瑞,何兵寿,胡楠. 基于一阶速度—胀缩—旋转方程的多分量联合逆时偏移[J]. 煤炭学报,2018,43(4): 1072-1082.LI Kairui,HE Bingshou,HU Nan. Multicomponent joint inverse time migration based on first order velo-city-dilatation-rotation equations[J]. Journal of China Coal Society,2018,43(4): 1072-1082.